In my use case, e is the result of a lengthy computation where at some point beyond my control the result was coerced into QQ[X]. In my particular case, I prefer the fraction field of ZZ[X] because the output is nicer.

Just a hack. An other one would be to get the lcm of the denominators of the coefficients appearing in the fraction. Explicitly. Here is an awful example, taken so to test if things work in a real mess.